Exponential and Logarithmic Functions 3.1
Exponential Functions and Their Graphs
3.2
Logarithmic Functions and Their Graphs
3.3
Properties of Logarithms
3.4
Exponential and Logarithmic Equations
3.5
Exponential and Logarithmic Models
3
In Mathematics Exponential functions involve a constant base and a variable exponent. The inverse of an exponential function is a logarithmic function.
Exponential and logarithmic functions are widely used in describing economic and physical phenomena such as compound interest, population growth, memory retention, and decay of radioactive material. For instance, a logarithmic function can be used to relate an animal’s weight and its lowest galloping speed. (See Exercise 95, page 242.)
Juniors Bildarchiv / Alamy
In Real Life
IN CAREERS There are many careers that use exponential and logarithmic functions. Several are listed below. • Astronomer Example 7, page 240
• Archeologist Example 3, page 258
• Psychologist Exercise 136, page 253
• Forensic Scientist Exercise 75, page 266
215
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216
Chapter 3
Exponential and Logarithmic Functions
3.1 EXPONENTIAL FUNCTIONS AND THEIR GRAPHS What you should learn • Recognize and evaluate exponential functions with base a. • Graph exponential functions and use the One-to-One Property. • Recognize, evaluate, and graph exponential functions with base e. • Use exponential functions to model and solve real-life problems.
Why you should learn it
So far, this text has dealt mainly with algebraic functions, which include polynomial functions and rational functions. In this chapter, you will study two types of nonalgebraic functions—exponential functions and logarithmic functions. These functions are examples of transcendental functions.
Definition of Exponential Function The exponential function f with base a is denoted by f x a x where a > 0, a 1, and x is any real number.
The base a 1 is excluded because it yields f x 1x 1. This is a constant function, not an exponential function. You have evaluated a x for integer and rational values of x. For example, you know that 43 64 and 412 2. However, to evaluate 4x for any real number x, you need to interpret forms with irrational exponents. For the purposes of this text, it is sufficient to think of a 2
(where 2 1.41421356)
as the number that has the successively closer approximations a1.4, a1.41, a1.414, a1.4142, a1.41421, . . . .
Example 1
Evaluating Exponential Functions
Use a calculator to evaluate each function at the indicated value of x. Function a. f x 2 x b. f x 2x c. f x 0.6x
Value x 3.1 x x 32
Solution >
Graphing Calculator Keystrokes ⴚ 3.1 ENTER 2 ⴚ ENTER 2 >
Function Value a. f 3.1 23.1 b. f 2 3 c. f 2 0.632
.6
>
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Exponential functions can be used to model and solve real-life problems. For instance, in Exercise 76 on page 226, an exponential function is used to model the concentration of a drug in the bloodstream.
Exponential Functions
3
ⴜ
2
ENTER
Display 0.1166291 0.1133147 0.4647580
Now try Exercise 7. When evaluating exponential functions with a calculator, remember to enclose fractional exponents in parentheses. Because the calculator follows the order of operations, parentheses are crucial in order to obtain the correct result.
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Section 3.1
Exponential Functions and Their Graphs
217
Graphs of Exponential Functions The graphs of all exponential functions have similar characteristics, as shown in Examples 2, 3, and 5.
Example 2
Graphs of y ⴝ a x
In the same coordinate plane, sketch the graph of each function. a. f x 2x You can review the techniques for sketching the graph of an equation in Section 1.2.
y
b. gx 4x
Solution The table below lists some values for each function, and Figure 3.1 shows the graphs of the two functions. Note that both graphs are increasing. Moreover, the graph of gx 4x is increasing more rapidly than the graph of f x 2x.
g(x) = 4x
16
x
3
2
1
0
1
2
14
2x
1 8
1 4
1 2
1
2
4
4x
1 64
1 16
1 4
1
4
16
12 10 8 6
Now try Exercise 17.
4
f(x) = 2x
2
x
−4 − 3 −2 −1 −2 FIGURE
1
2
3
4
The table in Example 2 was evaluated by hand. You could, of course, use a graphing utility to construct tables with even more values.
Example 3
3.1
G(x) = 4 −x
Graphs of y ⴝ a–x
In the same coordinate plane, sketch the graph of each function.
y
a. Fx 2x
16 14
b. Gx 4x
Solution
12
The table below lists some values for each function, and Figure 3.2 shows the graphs of the two functions. Note that both graphs are decreasing. Moreover, the graph of Gx 4x is decreasing more rapidly than the graph of Fx 2x.
10 8 6 4
F(x) =
− 4 −3 − 2 −1 −2 FIGURE
2
1
0
1
2
3
2x
4
2
1
1 2
1 4
1 8
4x
16
4
1
1 4
1 16
1 64
x
2 −x x
1
2
3
4
3.2
Now try Exercise 19. In Example 3, note that by using one of the properties of exponents, the functions F x 2x and Gx 4x can be rewritten with positive exponents. F x 2x
1 1 x 2 2
x
and Gx 4x
1 1 x 4 4
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x
218
Chapter 3
Exponential and Logarithmic Functions
Comparing the functions in Examples 2 and 3, observe that Fx 2x f x
and
Gx 4x gx.
Consequently, the graph of F is a reflection (in the y-axis) of the graph of f. The graphs of G and g have the same relationship. The graphs in Figures 3.1 and 3.2 are typical of the exponential functions y a x and y ax. They have one y-intercept and one horizontal asymptote (the x-axis), and they are continuous. The basic characteristics of these exponential functions are summarized in Figures 3.3 and 3.4. y
Notice that the range of an exponential function is 0, , which means that a x > 0 for all values of x.
y = ax (0, 1) x
FIGURE
3.3 y
y = a −x (0, 1) x
FIGURE
Graph of y a x, a > 1 • Domain: , • Range: 0, • y-intercept: 0, 1 • Increasing • x-axis is a horizontal asymptote ax → 0 as x→ . • Continuous
Graph of y ax, a > 1 • Domain: , • Range: 0, • y-intercept: 0, 1 • Decreasing • x-axis is a horizontal asymptote ax → 0 as x→ . • Continuous
3.4
From Figures 3.3 and 3.4, you can see that the graph of an exponential function is always increasing or always decreasing. As a result, the graphs pass the Horizontal Line Test, and therefore the functions are one-to-one functions. You can use the following One-to-One Property to solve simple exponential equations. For a > 0 and a 1, ax ay if and only if x y.
Example 4
Using the One-to-One Property
a. 9 3x1 32 3x1
Original equation 9 32
2x1 1x b.
1 x 2
One-to-One Property
One-to-One Property Solve for x.
8 ⇒ 2x 23 ⇒ x 3 Now try Exercise 51.
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Section 3.1
219
Exponential Functions and Their Graphs
In the following example, notice how the graph of y a x can be used to sketch the graphs of functions of the form f x b ± a xc.
Example 5 You can review the techniques for transforming the graph of a function in Section 1.7.
Transformations of Graphs of Exponential Functions
Each of the following graphs is a transformation of the graph of f x 3x. a. Because gx 3x1 f x 1, the graph of g can be obtained by shifting the graph of f one unit to the left, as shown in Figure 3.5. b. Because hx 3x 2 f x 2, the graph of h can be obtained by shifting the graph of f downward two units, as shown in Figure 3.6. c. Because kx 3x f x, the graph of k can be obtained by reflecting the graph of f in the x-axis, as shown in Figure 3.7. d. Because j x 3x f x, the graph of j can be obtained by reflecting the graph of f in the y-axis, as shown in Figure 3.8. y
y 2
3
g(x) =
3x + 1
f (x) =
3x 1
2 x
−2
1
−2 FIGURE
−1
f(x) = 3 x
h(x) = 3 x − 2 −2
1
3.5 Horizontal shift
FIGURE
3.6 Vertical shift
y
y
2 1
4 3
f(x) = 3 x x
−2
1 −1
2
k(x) = −3 x
−2 FIGURE
2
−1 x
−1
1
3.7 Reflection in x-axis
2
j(x) = 3 −x
f(x) = 3 x 1 x
−2 FIGURE
−1
1
2
3.8 Reflection in y-axis
Now try Exercise 23. Notice that the transformations in Figures 3.5, 3.7, and 3.8 keep the x-axis as a horizontal asymptote, but the transformation in Figure 3.6 yields a new horizontal asymptote of y 2. Also, be sure to note how the y-intercept is affected by each transformation.
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220
Chapter 3
Exponential and Logarithmic Functions
The Natural Base e y
In many applications, the most convenient choice for a base is the irrational number e 2.718281828 . . . .
3
(1, e)
This number is called the natural base. The function given by f x e x is called the natural exponential function. Its graph is shown in Figure 3.9. Be sure you see that for the exponential function f x e x, e is the constant 2.718281828 . . . , whereas x is the variable.
2
f(x) = e x
(− 1, e −1)
(0, 1)
Example 6
(− 2, e −2) −2 FIGURE
x
−1
1
Use a calculator to evaluate the function given by f x e x at each indicated value of x. a. b. c. d.
3.9
Evaluating the Natural Exponential Function
x 2 x 1 x 0.25 x 0.3
Solution y
a. b. c. d.
8
f(x) = 2e 0.24x
7 6 5
Function Value f 2 e2 f 1 e1 f 0.25 e0.25 f 0.3 e0.3
Graphing Calculator Keystrokes ex ⴚ 2 ENTER ex ⴚ 1 ENTER ex 0.25 ENTER ex ⴚ 0.3 ENTER
Display 0.1353353 0.3678794 1.2840254 0.7408182
Now try Exercise 33.
4 3
Example 7
Graphing Natural Exponential Functions
1 x
−4 −3 −2 −1 FIGURE
1
2
3
4
Sketch the graph of each natural exponential function. a. f x 2e0.24x b. gx 12e0.58x
3.10
Solution
y
To sketch these two graphs, you can use a graphing utility to construct a table of values, as shown below. After constructing the table, plot the points and connect them with smooth curves, as shown in Figures 3.10 and 3.11. Note that the graph in Figure 3.10 is increasing, whereas the graph in Figure 3.11 is decreasing.
8 7 6 5 4
2
g(x) = 12 e −0.58x
1 −4 −3 − 2 − 1 FIGURE
3.11
3
2
1
0
1
2
3
f x
0.974
1.238
1.573
2.000
2.542
3.232
4.109
gx
2.849
1.595
0.893
0.500
0.280
0.157
0.088
x
3
x 1
2
3
4
Now try Exercise 41.
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Section 3.1
Exponential Functions and Their Graphs
221
Applications One of the most familiar examples of exponential growth is an investment earning continuously compounded interest. Using exponential functions, you can develop a formula for interest compounded n times per year and show how it leads to continuous compounding. Suppose a principal P is invested at an annual interest rate r, compounded once per year. If the interest is added to the principal at the end of the year, the new balance P1 is P1 P Pr P1 r. This pattern of multiplying the previous principal by 1 r is then repeated each successive year, as shown below. Year 0 1 2 3 .. . t
Balance After Each Compounding PP P1 P1 r P2 P11 r P1 r1 r P1 r2 P3 P21 r P1 r21 r P1 r3 .. . Pt P1 rt
To accommodate more frequent (quarterly, monthly, or daily) compounding of interest, let n be the number of compoundings per year and let t be the number of years. Then the rate per compounding is rn and the account balance after t years is
AP 1
m
1
1 m
m
r n
. nt
If you let the number of compoundings n increase without bound, the process approaches what is called continuous compounding. In the formula for n compoundings per year, let m nr. This produces
r n
P 1
2.704813829
r mr
1,000
2.716923932
P 1
10,000
2.718145927
1 m
100,000
2.718268237
1,000,000
2.718280469
10,000,000
2.718281693
e
1
2
10
2.59374246
100
Amount (balance) with n compoundings per year
AP 1
P
1
nt
Amount with n compoundings per year
mrt
Substitute mr for n.
mrt
1 m
Simplify.
. m rt
Property of exponents
As m increases without bound, the table at the left shows that 1 1mm → e as m → . From this, you can conclude that the formula for continuous compounding is A Pert.
Substitute e for 1 1mm.
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Chapter 3
Exponential and Logarithmic Functions
WARNING / CAUTION Be sure you see that the annual interest rate must be written in decimal form. For instance, 6% should be written as 0.06.
Formulas for Compound Interest After t years, the balance A in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas.
1. For n compoundings per year: A P 1
r n
nt
2. For continuous compounding: A Pe rt
Example 8
Compound Interest
A total of $12,000 is invested at an annual interest rate of 9%. Find the balance after 5 years if it is compounded a. quarterly. b. monthly. c. continuously.
Solution a. For quarterly compounding, you have n 4. So, in 5 years at 9%, the balance is
AP 1
r n
nt
Formula for compound interest
12,000 1
0.09 4
4(5)
Substitute for P, r, n, and t.
$18,726.11.
Use a calculator.
b. For monthly compounding, you have n 12. So, in 5 years at 9%, the balance is
AP 1
r n
nt
12,000 1
Formula for compound interest
0.09 12
12(5)
$18,788.17.
Substitute for P, r, n, and t. Use a calculator.
c. For continuous compounding, the balance is A Pe rt
Formula for continuous compounding
12,000e0.09(5)
Substitute for P, r, and t.
$18,819.75.
Use a calculator.
Now try Exercise 59. In Example 8, note that continuous compounding yields more than quarterly or monthly compounding. This is typical of the two types of compounding. That is, for a given principal, interest rate, and time, continuous compounding will always yield a larger balance than compounding n times per year.
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Section 3.1
Example 9
223
Exponential Functions and Their Graphs
Radioactive Decay
The half-life of radioactive radium 226Ra is about 1599 years. That is, for a given amount of radium, half of the original amount will remain after 1599 years. After another 1599 years, one-quarter of the original amount will remain, and so on. Let y represent the mass, in grams, of a quantity of radium. The quantity present after t t1599 years, then, is y 2512 . a. What is the initial mass (when t 0)? b. How much of the initial mass is present after 2500 years?
Graphical Solution
Algebraic Solution
12 1 25 2
a. y 25
Use a graphing utility to graph y 2512
t1599
t1599
Write original equation.
a. Use the value feature or the zoom and trace features of the graphing utility to determine that when x 0, the value of y is 25, as shown in Figure 3.12. So, the initial mass is 25 grams. b. Use the value feature or the zoom and trace features of the graphing utility to determine that when x 2500, the value of y is about 8.46, as shown in Figure 3.13. So, about 8.46 grams is present after 2500 years.
01599
Substitute 0 for t.
25
Simplify.
So, the initial mass is 25 grams.
12 1 25 2
t1599
b. y 25
25
.
12
8.46
Write original equation.
30
30
25001599
Substitute 2500 for t. 1.563
Simplify. Use a calculator.
0
So, about 8.46 grams is present after 2500 years.
5000 0
FIGURE
0
5000 0
3.12
FIGURE
3.13
Now try Exercise 73.
CLASSROOM DISCUSSION Identifying Exponential Functions Which of the following functions generated the two tables below? Discuss how you were able to decide. What do these functions have in common? Are any of them the same? If so, explain why. b. f2x ⴝ 8 2
1 c. f3x ⴝ 2xⴚ3
e. f5x ⴝ 7 ⴙ 2x
f. f6x ⴝ 82x
1 x
a. f1x ⴝ 2xⴙ3
d. f4x ⴝ 12 ⴙ 7 x
x
1
0
1
2
3
x
2
1
0
1
2
gx
7.5
8
9
11
15
hx
32
16
8
4
2
Create two different exponential functions of the forms y ⴝ ab x and y ⴝ c x ⴙ d with y-intercepts of 0, ⴚ3.
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224
Chapter 3
3.1
Exponential and Logarithmic Functions
EXERCISES
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
VOCABULARY: Fill in the blanks. 1. 2. 3. 4.
Polynomial and rational functions are examples of ________ functions. Exponential and logarithmic functions are examples of nonalgebraic functions, also called ________ functions. You can use the ________ Property to solve simple exponential equations. The exponential function given by f x e x is called the ________ ________ function, and the base e is called the ________ base. 5. To find the amount A in an account after t years with principal P and an annual interest rate r compounded n times per year, you can use the formula ________. 6. To find the amount A in an account after t years with principal P and an annual interest rate r compounded continuously, you can use the formula ________.
SKILLS AND APPLICATIONS In Exercises 7–12, evaluate the function at the indicated value of x. Round your result to three decimal places. 7. 8. 9. 10. 11. 12.
Function
Value
f x 0.9 f x 2.3x f x 5x 5x f x 23 g x 50002x f x 2001.212x
x 1.4 3 x2 x 3 x 10 x 1.5 x 24
x
17. f x 12 19. f x 6x 21. f x 2 x1 x
y
6
4
4
−2
x 2
−2
4
−2
y
(c)
−4
−2
x 2
6
4
4
2
13. f x 2 15. f x 2x x
4
6
(0, 1) −4
−2
30. y 3 x
32. y 4x1 2
In Exercises 33–38, evaluate the function at the indicated value of x. Round your result to three decimal places.
2 4
x
In Exercises 29–32, use a graphing utility to graph the exponential function. 29. y 2x 31. y 3x2 1
y
6
−2
f x 3 x, gx 3 x 1 f x 4 x, gx 4 x3 f x 2 x, gx 3 2 x f x 10 x, gx 10 x3
2
(d)
x
23. 24. 25. 26.
x
−2
(0, 2)
x
27. f x 72 , gx 72 28. f x 0.3 x, gx 0.3 x 5
(0, 14 (
(0, 1) −4
y
(b)
6
18. f x 12 20. f x 6 x 22. f x 4 x3 3
In Exercises 23–28, use the graph of f to describe the transformation that yields the graph of g.
In Exercises 13–16, match the exponential function with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a)
In Exercises 17–22, use a graphing utility to construct a table of values for the function. Then sketch the graph of the function.
2
−2
14. f x 2 1 16. f x 2x2 x
x 4
Function 33. hx ex 34. f x e x 35. f x 2e5x 36. f x 1.5e x2 37. f x 5000e0.06x 38. f x 250e0.05x
Value x 34 x 3.2 x 10 x 240 x6 x 20
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Section 3.1
In Exercises 39–44, use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. 39. f x e x 41. f x 3e x4 43. f x 2e x2 4
40. f x e x 42. f x 2e0.5x 44. f x 2 e x5
In Exercises 45–50, use a graphing utility to graph the exponential function. 45. y 1.085x 47. st 2e0.12t 49. gx 1 ex
46. y 1.085x 48. st 3e0.2t 50. hx e x2
In Exercises 51–58, use the One-to-One Property to solve the equation for x. 51. 3x1 27
52. 2x3 16
53. 32 55. e3x2 e3 2 57. ex 3 e2x 1 x 2
1 54. 5x2 125 56. e2x1 e4 2 58. ex 6 e5x
COMPOUND INTEREST In Exercises 59–62, complete the table to determine the balance A for P dollars invested at rate r for t years and compounded n times per year. n
1
2
4
12
365
Continuous
A 59. 60. 61. 62.
P $1500, r 2%, t 10 years P $2500, r 3.5%, t 10 years P $2500, r 4%, t 20 years P $1000, r 6%, t 40 years
COMPOUND INTEREST In Exercises 63–66, complete the table to determine the balance A for $12,000 invested at rate r for t years, compounded continuously. t
10
20
30
40
50
A 63. r 4% 65. r 6.5%
64. r 6% 66. r 3.5%
67. TRUST FUND On the day of a child’s birth, a deposit of $30,000 is made in a trust fund that pays 5% interest, compounded continuously. Determine the balance in this account on the child’s 25th birthday.
Exponential Functions and Their Graphs
225
68. TRUST FUND A deposit of $5000 is made in a trust fund that pays 7.5% interest, compounded continuously. It is specified that the balance will be given to the college from which the donor graduated after the money has earned interest for 50 years. How much will the college receive? 69. INFLATION If the annual rate of inflation averages 4% over the next 10 years, the approximate costs C of goods or services during any year in that decade will be modeled by Ct P1.04 t, where t is the time in years and P is the present cost. The price of an oil change for your car is presently $23.95. Estimate the price 10 years from now. 70. COMPUTER VIRUS The number V of computers infected by a computer virus increases according to the model Vt 100e4.6052t, where t is the time in hours. Find the number of computers infected after (a) 1 hour, (b) 1.5 hours, and (c) 2 hours. 71. POPULATION GROWTH The projected populations of California for the years 2015 through 2030 can be modeled by P 34.696e0.0098t, where P is the population (in millions) and t is the time (in years), with t 15 corresponding to 2015. (Source: U.S. Census Bureau) (a) Use a graphing utility to graph the function for the years 2015 through 2030. (b) Use the table feature of a graphing utility to create a table of values for the same time period as in part (a). (c) According to the model, when will the population of California exceed 50 million? 72. POPULATION The populations P (in millions) of Italy from 1990 through 2008 can be approximated by the model P 56.8e0.0015t, where t represents the year, with t 0 corresponding to 1990. (Source: U.S. Census Bureau, International Data Base) (a) According to the model, is the population of Italy increasing or decreasing? Explain. (b) Find the populations of Italy in 2000 and 2008. (c) Use the model to predict the populations of Italy in 2015 and 2020. 73. RADIOACTIVE DECAY Let Q represent a mass of radioactive plutonium 239Pu (in grams), whose halflife is 24,100 years. The quantity of plutonium present t24,100 after t years is Q 1612 . (a) Determine the initial quantity (when t 0). (b) Determine the quantity present after 75,000 years. (c) Use a graphing utility to graph the function over the interval t 0 to t 150,000.
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226
Chapter 3
Exponential and Logarithmic Functions
74. RADIOACTIVE DECAY Let Q represent a mass of carbon 14 14C (in grams), whose half-life is 5715 years. The quantity of carbon 14 present after t years is 1 t5715 Q 102 . (a) Determine the initial quantity (when t 0). (b) Determine the quantity present after 2000 years. (c) Sketch the graph of this function over the interval t 0 to t 10,000. 75. DEPRECIATION After t years, the value of a wheelchair conversion van that originally cost $30,500 depreciates so that each year it is worth 78 of its value for the previous year. (a) Find a model for Vt, the value of the van after t years. (b) Determine the value of the van 4 years after it was purchased. 76. DRUG CONCENTRATION Immediately following an injection, the concentration of a drug in the bloodstream is 300 milligrams per milliliter. After t hours, the concentration is 75% of the level of the previous hour. (a) Find a model for Ct, the concentration of the drug after t hours. (b) Determine the concentration of the drug after 8 hours.
84. Use a graphing utility to graph each function. Use the graph to find where the function is increasing and decreasing, and approximate any relative maximum or minimum values. (a) f x x 2ex (b) gx x23x 85. GRAPHICAL ANALYSIS Use a graphing utility to graph y1 1 1xx and y2 e in the same viewing window. Using the trace feature, explain what happens to the graph of y1 as x increases. 86. GRAPHICAL ANALYSIS Use a graphing utility to graph
f x 1
0.5 x
x
gx e0.5
and
in the same viewing window. What is the relationship between f and g as x increases and decreases without bound? 87. GRAPHICAL ANALYSIS Use a graphing utility to graph each pair of functions in the same viewing window. Describe any similarities and differences in the graphs. (a) y1 2x, y2 x2 (b) y1 3x, y2 x3 88. THINK ABOUT IT Which functions are exponential? (a) 3x (b) 3x 2 (c) 3x (d) 2x 89. COMPOUND INTEREST Use the formula
r n
nt
EXPLORATION
AP 1
TRUE OR FALSE? In Exercises 77 and 78, determine whether the statement is true or false. Justify your answer.
to calculate the balance of an account when P $3000, r 6%, and t 10 years, and compounding is done (a) by the day, (b) by the hour, (c) by the minute, and (d) by the second. Does increasing the number of compoundings per year result in unlimited growth of the balance of the account? Explain.
77. The line y 2 is an asymptote for the graph of f x 10 x 2. 271,801 78. e 99,990 THINK ABOUT IT In Exercises 79– 82, use properties of exponents to determine which functions (if any) are the same. 79. f x 3x2 gx 3x 9 hx 193x 81. f x 164x x2 gx 14 hx 1622x
80. f x 4x 12 gx 22x6 hx 644x 82. f x ex 3 gx e3x hx e x3
83. Graph the functions given by y 3x and y 4x and use the graphs to solve each inequality. (a) 4x < 3x (b) 4x > 3x
90. CAPSTONE The figure shows the graphs of y 2x, y ex, y 10x, y 2x, y ex, and y 10x. Match each function with its graph. [The graphs are labeled (a) through (f).] Explain your reasoning. y
c 10 b
d
8
e
6
a −2 −1
f x 1
2
PROJECT: POPULATION PER SQUARE MILE To work an extended application analyzing the population per square mile of the United States, visit this text’s website at academic.cengage.com. (Data Source: U.S. Census Bureau)
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